Question

Eliminate $m$ from the equations $y+mx=a(1+m),my-x=a(1-m)$.

Answer 1

$y+mx=a(1+m)$ ..... $\left(i\right)$

$my-x=a(1-m)$ ..... $\left(ii\right)$

Taking square of the equations and add, we get
$(y+mx{)}^2+(my-x{)}^2=\left(a\right(1+m){)}^2+(a(1-m){)}^2$
$m^2(x^2+y^2)+(x^2+y^2)=a^2\left(2\right(1+m^2\left)\right)$
$(x^2+y^2)(m^2+1)=2a^2(1+m^2)$
Therefore, the eliminated equation is
$x^2+y^2=2a^2$